Metzler matrix

In mathematics, especially linear algebra, a matrix is called Metzler, quasipositive (or quasi-positive) or essentially nonnegative if all of its elements are non-negative except for those on the main diagonal, which are unconstrained. That is, a Metzler matrix is any matrix A which satisfies

${\displaystyle A=(a_{ij});\quad a_{ij}\geq 0,\quad i\neq j.}$

Metzler matrices are also sometimes referred to as ${\displaystyle Z^{(-)}}$-matrices, as a Z-matrix is equivalent to a negated quasipositive matrix.

The exponential of a Metzler (or quasipositive) matrix is a nonnegative matrix because of the corresponding property for the exponential of a nonnegative matrix. This is natural, once one observes that the generator matrices of continuous-time Markov chains are always Metzler matrices, and that probability distributions are always non-negative.

A Metzler matrix has an eigenvector in the nonnegative orthant because of the corresponding property for nonnegative matrices.

• Perron–Frobenius theorem

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